Let \(\mathbf {U}\)  be a quantum group associated to a quiver \(Q=(I,H)\)  and \(\mathbf {U}^-\)  be its negative part. The algebra \(\mathbf {U}^-\)  is defined by the generators \(F_i\)  for \(i\in I\)  and quantum Serre relations.
In [1]}, Ringel introduced the twisted Ringel-Hall algebra \(H^{\ast }_q(Q)\)  and proved that the composition subalgebra of \(H^{\ast }_q(Q)\)  is isomorphic to a specilization of \(\mathbf {U}^-\) . In [2]}, Green introduced the comultiplication and showed that the comultiplication is an algebra homomorphism by using Green's formula.
